# 61 Einstein’s Simple Arguments Establishing Special Relativity

*Before launching into
the elegant mathematical analysis of special relativity, it is perhaps illuminating
to see how Einstein himself developed it, using very elementary mathematics,
and a few simple thought experiments.*

## Einstein Puts Maxwell and Galileo Together

When Maxwell discovered that his electromagnetic equations predicted wave solutions, with a (predicted) wave speed equal to that measured for light, he concluded that light was an electromagnetic wave. But what was waving? Of course, he was familiar with the mathematically identical equations for sound waves in air (and also in solids, where transverse oscillations are possible), so he naturally assumed space to be filled with some appropriate medium, dubbed the “ether”. He (and others) went to great lengths to construct mechanical models of this ether, and experimentalists spent years trying to detect its motion by very accurate measurements of the speed of light. They believed that, like sound in air, an ether wind (such as caused by the Earth’s motion) would carry the signal and therefore change its measured velocity.

But nothing succeeded. The ether proved elusive. Then Einstein came along. He referred back to Galileo’s formulation of invariance: the Laws of Physics are the same in any initial frame of reference. Galileo’s example was that inside a ship moving at constant velocity, no experiment (such as throwing or dropping a ball) could detect the ship’s motion. Einstein simply generalized Galileo’s idea to include the newly discovered laws: Maxwell’s equations. But those equations make a specific prediction of the speed of light, $c=3\times {10}^{8}$ meters per second. Therefore, the speed of light must be equal to this number in all inertial frames! But what about addition of velocities? The simple formula for parallel velocities, $v={v}_{1}+{v}_{2},$ must be wrong… .

Einstein began with the constancy of the speed of light and, with very simple arguments, showed how this led to time dilation, length contraction, and $E=m{c}^{2}.$ I think it’s worthwhile looking through these arguments, from my Modern Physics notes, before becoming immersed in the formalism. Here they are. (Needless to say, all this is now completely confirmed experimentally.)

## A Few (Easy!) Modern Physics Lectures

Using a simple clock to establish time dilation and length contraction.

Clocks synchronized in one frame won’t in general be synchronized viewed from another frame.

Einstein’s elementary derivation of the Lorentz transformation.

How can two observers in different inertial frames each see the other’s clock as running slow? This is worth reading through.

Elementary derivation of the Addition of Velocities formula.

Mass and energy: Einstein’s Box.

Lorentz transformations of Energy and Momentum.